Find the minimum and maximum values of $f(x) = \frac{x^2 + 1}{x}$ on $[-2,0)$.

We are tasked with finding the minimum and maximum values of the function $ f(x) = \frac{x^2 + 1}{x} $ on the interval $ [-2, 0) $.

### Step 1: Rewrite the Function
First, simplify $ f(x) $:

\[
f(x) = \frac{x^2 + 1}{x} = x + \frac{1}{x}
\]

### Step 2: Compute the Derivative
To find the critical points, we first need to compute the derivative of $ f(x) $.

\[
f'(x) = \frac{d}{dx}\left( x + \frac{1}{x} \right) = 1 - \frac{1}{x^2}
\]

### Step 3: Solve for Critical Points
Set $ f'(x) = 0 $ to find the critical points:

\[
1 - \frac{1}{x^2} = 0
\]

\[
\frac{1}{x^2} = 1 \quad \Rightarrow \quad x^2 = 1 \quad \Rightarrow \quad x = \pm 1
\]

However, since $ x $ is restricted to the interval $ [-2, 0) $, only $ x = -1 $ is valid.

### Step 4: Evaluate the Function at Critical Points and Endpoints
Now, evaluate $ f(x) $ at the critical point and at the endpoints of the interval.

- **At $ x = -1 $:**

\[
f(-1) = -1 + \frac{1}{-1} = -1 - 1 = -2
\]

- **At $ x = -2 $:**

\[
f(-2) = -2 + \frac{1}{-2} = -2 - \frac{1}{2} = -\frac{5}{2}
\]

- **At $ x \to 0^- $ (as $ x $ approaches 0 from the left):**

\[
f(x) = x + \frac{1}{x} \quad \text{approaches} \quad -\infty
\]
as $ \frac{1}{x} $ tends to $ -\infty $.

### Step 5: Determine Minimum and Maximum Values
From the evaluations:
- The **maximum value** is $ f(-1) = -2 $.
- The function decreases without bound as $ x \to 0^- $, so the **minimum value** does not exist (the function tends to $ -\infty $).

### Conclusion:
- The maximum value of $ f(x) $ on $ [-2, 0) $ is $ -2 $.
- The function has no minimum value as it tends to $ -\infty $ as $ x \to 0^- $.

Would you like further details on this solution?

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Here are 5 related questions:
1. How would the problem change if the interval included 0?
2. What is the behavior of $ f(x) $ on a different interval, such as $ (-\infty, 0) $?
3. Can you explain the role of critical points in determining extremum values?
4. What are the conditions for a function to have absolute extrema?
5. How does asymptotic behavior influence the analysis of functions near boundary points?

**Tip:** Always analyze the behavior of functions near the boundaries when dealing with open intervals to understand how the function behaves as it approaches those points.

Explore how to find the minimum and maximum values of the function f(x) = (x^2 + 1)/x on the interval [-2,0). This problem involves key calculus concepts such as derivatives, critical points, and limits, suitable for high school or early college students.

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